In this article, we establish an important property about the growth of sequences in the dual space of Cesàro sequence spaces. As a consequence of this fact, we calculate the measure of noncompactness or the essential norm of the multiplication operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> acting on Cesàro sequence spaces <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>c</mml:mi><mml:mi>e</mml:mi><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math>.